Mathquery

Lioville's Theorem :- Statement :       If a function f(z) is analytic for all limit values of z and is bounded then f(z) is constant . Proof :        Let z₁ , z₂ be any two point of the z-plane the contour C to be a large circle of radius R centred at origin and containing the point z₁ , z₂ . Therefore |r₁|<R and |z₂|<R . Also as f(z) is bounded therefore ∃ a positive M such that f|(z)|≤M for all z.      By Cauchy's Integral formula we have       f(z₁) = 1/2πi ∫ f(z) dz /z-z₁                           c      f(z₂) = 1/2πi ∫ f(z) dz /z-z₂                           c ∴ f(z₁) - f(z₂) = 1/2πi ∫ f(z) dz /z-z₁                                     c                        - 1/2πi ∫ f(z) dz /z-z₂                                     c                  = 1/2πi ∫ (z₁-z₂) f(z) dz /(z-z₁)(z-z₂)                               c |f(z₁)-f(z₂)|              =   |1/2πi ∫(z₁-z₂)f(z) dz /(z-z₁)(z-z₂)|